LINEAR ALGEBRA AND GEOMETRY Ps - Z

The course deals with the following subjects

1) Linear Algebra: resolution of linear systems, study of linear maps, eigenvalues and eigenvectors for endomorphisms;

2) Geometry: linear geometry in the plane and in 3-dimensional space (lines and planes), study of conics and quadrics.

The student will be able to apply these notions and methods to the resolution of concrete problems of linear algebra and analytical geometry concerning the study of the simplest geometrical objects in the plane and in the space.

By studying linear algebra and geometry, the students will address various theoretical aspects of the topics addressed and will test themselves through exercises. They will learn how to use a correct language for a scientific communication.

Lectures, active learning and classroom participation. Exercises.

Linear Algebra:

1. Generalities on set theory and operations. Maps between sets, image and inverse image, injective and surjective maps, bijective maps. Sets with operation, gropus, rings, fields.
2. Vectors in the ordinary space. Sum of vectors, product of a number and a vector. Scalar product, vector product. Components of vectors and operations with components.
3. Complex numbers, operations and properties. Algebraic and trigonometric form of complex numbers. De Moivre formula. nth root of complex numbers.
4. Vector spaces and properties. Examples. Subspaces. Intersection, union and sum of subspaces. Linear independence. Generators. Base of a vector space, completion of a base. Steinitz Lemma*, dimension of a vector space. Grassmann formula*. Direct sum.
5. Generalities on matrices. Rank. Reduced matrix and reduction of a matrix. Product of matrices. Linear systems. Rouchè-Capelli theorem. Solutions of linear systems. Homogeneous systems and space of solutions.
6. Determinants and properties. Laplace theorems*. Inverse of a square matrix. Binet theorem*. Cramer thoerem. Kronecker theorem*.
7. Linear maps and properties. Kernel and image. Injective and surjcetive maps. Isomorphisms. L(V,W) and isomomorphism with k^{m,n}. Study of a linear map. Base change.
8. Eigenvalues, eigenvectors and eigenspaces of an endomorphism. Characteristic polynomial. Dimension of eigenspaces. Independence of eigenvectors. Simple endomorphisms and diagonalization of matrices.

Geometry

1. Linear geometry on the plane. Cartesian coordinates and homogeneous coordinates. Lines and their equations. Intersection of lines. Angular coefficient. Distances. Pencils of lines.
2. Linear geometry in the space. Cartesian coordinates and homogeneous coordinates. Planes and their equation. Lines and their representation. Ideal elements. Angular properties of lines and planes. Distances. éencils of planes.
3. Change of coordinates in the plane, rotations and translations. Conics and associated matrices, ortogonal invariants. Reduced equations, reduction of a conic in canonic form. Classification of irreducible concis. Study of equations in canonic form. Circle. Tangent lines. Pencils of conics.
4. Quadrics in the space and associated matrices. Irreducible concis. Vertices and dengerate quadrics. Cones and cylinders. Reduced equations, reduction in canonic form. Classification of non degenerate quadrics. Sections of quadrics with lines and planes. Lines and tangent planes.

The prooves of the theorem signed with * can be ometted.

1. S. Giuffrida, A. Ragusa: Corso di Algebra Lineare con Esercizi Svolti. Il Cigno Galileo Galilei, Roma, 1998.
2. Lezioni di Geometria. Spazio Libri, Catania, 2000. Download disponibile gratuitamente al link http://www.giuseppepaxia.it/Prof_Paxia/Home_files/px.pdf